SSC/Structure/BiPolytropes/Analytic51/Pt3

BiPolytrope with n_c = 5 and n_e = 1 (Pt 3)

Part I:  (nc,ne) = (5,1) BiPolytrope

Part II:  Example Models

Part III:  Limiting Mass

Part IV:  Free Energy

Limiting Mass

Background

As early as 1941, Chandraskhar and his collaborators realized that the shape of the model sequence in a $\nu$ versus $q$ diagram, as displayed in Figure 2 above, implies that equilibrium structures can exist only if the fractional core mass lies below some limiting value. This realization is documented, for example, by the following excerpt from §5 of 📚 L. R. Henrich & S. Chandrasekhar (1941, ApJ, Vol. 94, pp. 525 - 536).

Text excerpt from §5 (pp. 532 - 533) of L. R. Henrich & S. Chandrasekhar (1941) Stellar Models with Isothermal Cores The Astrophysical Journal, Vol. 94, pp. 525 - 536

"… at a fixed central temperature, the fraction of the total mass, $\nu$, contained in the core increases slowly at first and soon very rapidly as $q$ approaches $q_{\mathrm{m}\mathrm{a}\mathrm{x}}$. However, this increase of $\nu$ does not continue indefinitely; $\nu$ soon attains a maximum value ${\nu }_{\mathrm{m}\mathrm{a}\mathrm{x}}$. There exists, therefore, an upper limit to the mass which can be contained in the isothermal core."

Given that our bipolytropic sequence has been defined analytically, it may be possible to analytically determine the limiting core mass of our model. In order to accomplish this, we need to identify the point along the sequence — in particular, the value of the dimensionless interface location — at which $d\nu /dq=0$ or, equivalently, $d\nu /d{\xi }_i=0$.

Before carrying out the desired differentiations, we will find it useful to rewrite the relevant expressions in terms of the parameters,

${\ell }_i\equiv \frac{{\xi }_i}{\sqrt{3}};$       and       $m_3\equiv 3\left(\frac{{\mu }_e}{{\mu }_c}\right).$

We obtain,

${\eta }_i$	$=$	$m_3\left(\frac{{\ell }_i}{1+{\ell }_i^2}\right);$
${\mathrm{\Lambda }}_i$	$=$	$\frac{1}{m_3{\ell }_i}[1+(1-m_3){\ell }_i^2]\Rightarrow$   Believe it or not …   $(1+{\mathrm{\Lambda }}^2)=\frac{(1+{\ell }_i^2)}{m_3^2{\ell }_i^2}[1+(1-m_3{)}^2{\ell }_i^2];$
$A$	$=$	$[\frac{1+(1-m_3{)}^2{\ell }_i^2}{1+{\ell }_i^2}{]}^{1/2};$
$(\frac{\pi }{2}{)}^{1/2}\frac{M_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{9}$	$=$	$\frac{{\ell }_i^3}{(1+{\ell }_i^2{)}^{3/2}};$
$(\frac{\pi }{2}{)}^{1/2}\frac{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{9}$	$=$	$\frac{1}{m_3^2}[1+(1-m_3{)}^2{\ell }_i^2{]}^{1/2}\left\{\right(\frac{\pi }{2}+{\tan }^{-1}{\mathrm{\Lambda }}_i)+m_3{\ell }_i(1+{\ell }_i^2{)}^{-1}\}.$

Hence,

$\nu \equiv \frac{M_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}=(m_3^2{\ell }_i^3)(1+{\ell }_i^2{)}^{-1/2}[1+(1-m_3{)}^2{\ell }_i^2{]}^{-1/2}[m_3{\ell }_i+(1+{\ell }_i^2)(\frac{\pi }{2}+{\tan }^{-1}{\mathrm{\Lambda }}_i){]}^{-1}$

An interesting limiting case is $m_3=1$, in which case, $\nu$ $=$ $({\ell }_i^3)(1+{\ell }_i^2{)}^{-1/2}[{\ell }_i+(1+{\ell }_i^2)(\frac{\pi }{2}+{\tan }^{-1}(\frac{1}{{\ell }_i}\left)\right){]}^{-1},$ and the maximum value of $\nu$ along this sequence arises when ${\ell }_i\to \mathrm{\infty }$, in which case, $\nu$ $\to$ ${\ell }_i^2[{\ell }_i+(1+{\ell }_i^2)(\frac{\pi }{2}){]}^{-1}\to \frac{2}{\pi }.$

The condition, $d\nu /d{\xi }_i=0$, also will be satisfied if the condition,

$\frac{d\ln \nu }{d\ln {\ell }_i}=0,$

is met.

Derivation

My manual derivation gives,

$(1+{\ell }_i^2)[\frac{\pi }{2}+{\tan }^{-1}{\mathrm{\Lambda }}_i]\{3-\frac{(1-m_3{)}^2{\ell }_i^2(1+{\ell }_i^2)}{[1+(1-m_3{)}^2{\ell }_i^2]}\}$

$=$

$(1+{\ell }_i^2)\frac{\partial {\tan }^{-1}{\mathrm{\Lambda }}_i}{\partial \ln {\ell }_i}-m_3{\ell }_i\{{\ell }_i^2+2-\frac{(1-m_3{)}^2{\ell }_i^2(1+{\ell }_i^2)}{[1+(1-m_3{)}^2{\ell }_i^2]}\}$

where,

$\frac{\partial {\tan }^{-1}{\mathrm{\Lambda }}_i}{\partial \ln {\ell }_i}=\frac{[(1-m_3){\ell }_i^2-1]}{m_3{\ell }_i(1+{\mathrm{\Lambda }}_i^2)}=\frac{m_3{\ell }_i[(1-m_3){\ell }_i^2-1]}{(1+{\ell }_i^2)[1+(1-m_3{)}^2{\ell }_i^2]}.$

Upon rearrangement, this gives,

$(1+{\ell }_i^2)[\frac{\pi }{2}+{\tan }^{-1}{\mathrm{\Lambda }}_i]\{3[1+(1-m_3{)}^2{\ell }_i^2]-(1-m_3{)}^2{\ell }_i^2(1+{\ell }_i^2)\}$

$=$

$m_3{\ell }_i\{[(1-m_3){\ell }_i^2-1]-({\ell }_i^2+2)[1+(1-m_3{)}^2{\ell }_i^2]+(1-m_3{)}^2{\ell }_i^2(1+{\ell }_i^2)\},$

and further simplification [completed on 19 May 2013] gives,

$\underset{\mathrm{L}\mathrm{H}\mathrm{S}}{\underbrace{(\frac{\pi }{2}+{\tan }^{-1}{\mathrm{\Lambda }}_i)(1+{\ell }_i^2)[3+(1-m_3{)}^2(2-{\ell }_i^2){\ell }_i^2]}}$

$=$

$\underset{\mathrm{R}\mathrm{H}\mathrm{S}}{\underbrace{m_3{\ell }_i[(1-m_3){\ell }_i^4-(m_3^2-m_3+2){\ell }_i^2-3]}}.$

Maximum Fractional Core Mass, $\nu =M_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}/M_{\mathrm{t}\mathrm{o}\mathrm{t}}$ (solid green circular markers) for Equilibrium Sequences having Various Values of ${\mu }_e/{\mu }_c$
$\frac{{\mu }_e}{{\mu }_c}$	${\xi }_i$	${\theta }_i$	${\eta }_i$	${\mathrm{\Lambda }}_i$	$A$	${\eta }_s$	LHS	RHS	$q\equiv \frac{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{R}$	$\nu \equiv \frac{M_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}$
$\frac{1}{3}$	$\mathrm{\infty }$	---	---	---	---	---	---	---	0.0	$\frac{2}{\pi }$
0.33	24.00496	0.0719668	0.0710624	0.2128753	0.0726547	1.8516032	-223.8157	-223.8159	0.038378833	0.52024552
0.316943	10.744571	0.1591479	0.1493938	0.4903393	0.1663869	2.1760793	-31.55254	-31.55254	0.068652714	0.382383875
0.3090	8.8301772	0.1924833	0.1750954	0.6130669	0.2053811	2.2958639	-18.47809	-18.47808	0.076265588	0.331475715
$\frac{1}{4}$	4.9379256	0.3309933	0.2342522	1.4179907	0.4064595	2.761622	-2.601255	-2.601257	0.084824137	0.139370157
Recall that, ${\ell }_i\equiv \frac{{\xi }_i}{\sqrt{3}};$       and       $m_3\equiv 3\left(\frac{{\mu }_e}{{\mu }_c}\right).$

Limit when m₃ = 0

It is instructive to examine the root of this equation in the limit where $m_3=0$ — that is, when ${\mu }_e/{\mu }_c=0$. First, we note that,

${\mathrm{\Lambda }}_i{|}_{m_3\to 0}=\{\frac{1}{m_3{\ell }_i}[1+(1-m_3){\ell }_i^2]{\}}_{m_3\to 0}=\mathrm{\infty }.$

Hence,

$[{\tan }^{-1}{\mathrm{\Lambda }}_i{]}_{m_3\to 0}=\frac{\pi }{2},$

and the limiting relation becomes,

$\pi (1+{\ell }_i^2)[3+(2-{\ell }_i^2){\ell }_i^2]=0,$

or, more simply,

${\ell }_i^4-2{\ell }_i^2-3=0.$

The real root is,

${\ell }_i^2=\frac{1}{2}[2+\sqrt{4+12}]=3\Rightarrow {\xi }_i=3.$

For ${\xi }_i=3$, the radius of the core, the mass of the core, and the pressure at the edge of the core are, respectively,

$r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}$	$=$	$(\frac{3^3}{2\pi }{)}^{1/2}$	$\Rightarrow$	$r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$	$=$	$\left(\frac{3^3}{2\pi }{)}^{1/2}\right[\frac{K_c^{1/2}}{G^{1/2}{\rho }_0^{2/5}}];$
$M_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}$	$=$	$(\frac{3^7}{2^5\pi }{)}^{1/2}$	$\Rightarrow$	$M_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$	$=$	$\left(\frac{3^7}{2^5\pi }{)}^{1/2}\right[\frac{K_c^{3/2}}{G^{3/2}{\rho }_0^{1/5}}];$
$P_i^{*}$	$=$	$2^{-6}$	$\Rightarrow$	$P_i$	$=$	$2^{-6}[K_c{\rho }_0^{6/5}].$

If we invert the middle expression to obtain ${\rho }_0$ in terms of $M_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$, specifically,

${\rho }_0^{1/5}=\left(\frac{3^7}{2^5\pi }{)}^{1/2}\right[\frac{K_c^{3/2}}{G^{3/2}{M}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}],$

then we can rewrite $r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$ and $P_i$ in terms of, respectively, the reference radius, $R_{\mathrm{r}\mathrm{f}}$, and reference pressure, $P_{\mathrm{r}\mathrm{f}}$, as defined in our discussion of isolated $n=5$ polytropes embedded in an external medium. Specifically, we obtain,

$r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$	$=$	$\left(\frac{3^3}{2\pi }{)}^{1/2}\right[\frac{K_c^{1/2}}{G^{1/2}}\left]\right(\frac{3^7}{2^5\pi }{)}^{-1}[\frac{K_c^{3/2}}{G^{3/2}{M}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{]}^{-2}$	$=$	$\left(\frac{2^9\pi }{3^{11}}{)}^{1/2}\right[\frac{G^{5/2}{M}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^2}{K_c^{5/2}}]$	$=$	$\left(\frac{2^9\pi }{3^{11}}{)}^{1/2}\frac{3^3}{2^6}\right(\frac{5^5}{\pi }{)}^{1/2}{R}_{\mathrm{r}\mathrm{f}}{|}_{n=5}$	$=$	$(\frac{5^5}{2^3\cdot 3^5}{)}^{1/2}{R}_{\mathrm{r}\mathrm{f}}{|}_{n=5}$
$P_i$	$=$	$2^{-6}[K_c]\left(\frac{3^7}{2^5\pi }{)}^3\right[\frac{K_c^{3/2}}{G^{3/2}{M}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{]}^6$	$=$	$\left(\frac{3^7}{2^7\pi }{)}^3\right[\frac{K_c^{10}}{G^9{M}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^6}]$	$=$	$\left(\frac{3^7}{2^7\pi }{)}^3\right(\frac{2^{26}{\pi }^3}{3^{12}{5}^9})P_{\mathrm{r}\mathrm{f}}{|}_{n=5}$	$=$	$\left(\frac{2^5\cdot 3^9}{5^9}\right)P_{\mathrm{r}\mathrm{f}}{|}_{n=5}$

[26 May 2013 with further elaboration on 28 May 2013] This is the same result that was obtained when we embedded an isolated $n=5$ polytrope in an external medium. Apparently, therefore, the physics that leads to the mass limit for a Bonnor-Ebert sphere is the same physics that sets the 📚 Schönberg & Chandrasekhar (1942) mass limit.

Derivation by Eggleton, Faulkner, and Cannon (1998)

The analytically prescribable sequence of bipolytropic models having $(n_c,n_e)=(5,1)$ displays an interesting behavior that extends beyond identification of a Schönberg-Chandrasekhar-like mass limit. After reaching a maximum value of $q$ but before reaching the maximum value of $\nu$, the sequence bends back on itself. This means that, even though the fraction of mass enclosed in the core is steadily increasing, the total radius of the configuration is increasing faster than the radius of the core. Qualitatively, at least, this mimics the behavior exhibited by normal stars as they evolve off the main sequence and up the red giant branch.

As I pondered whether or not to probe this analogy in more depth, I recalled — even dating back to my years as a graduate student at Lick Observatory — hearing John Faulkner profess that he finally understood why stars become red giants. I also recalled the following passage from [HK94]'s textbook on Stellar Interiors:

Excerpt from §2.3, p. 55 of [HK94]

"The enormous increase in radius that accompanies hydrogen exhaustion moves the star into the red giant region of the H-R diagram. While the transition to red giant dimensions is a fundamental result of all evolutionary calculations, a convincing yet intuitively satisfactory explanation of this dramatic transformation has not been formulated. Our discussion of this phenomenon follows that of Iben and Renzini (1984) although we must state that it is not the whole story."† _____________ †"Other attempts include: 📚 Eggleton & Faulkner (1981); Weiss (1983); Yahil & Van den Horn (1985); Applegate (1988); Whitworth (1989); Renzini et al. (1992). Bhaskar & Nigam (1991) use an interesting set of dimensional arguments plus notions from polytrope theory. We suspect the answers may lie in their paper but someone has yet to come along and translate the mathematics into an easily comprehensible physical picture."

While examining the set of authors who more recently have cited the work by 📚 Eggleton & Faulkner (1981), I discovered a paper by 📚 P. P. Eggleton, J. Faulkner, & R. C. Cannon (1998, MNRAS, Vol. 298, issue 3, pp. 831 - 834) with the following abstract:

P. P. Eggleton, J. Faulkner, & R. C. Cannon (1998) A Small Contribution to the Giant Problem Monthly Notices of the Royal Astronomical Society, Vol. 298, issue 3, pp. 831 - 834 Abstract:  "We present a simple analytic model of a composite polytropic star, which exhibits a limiting Schönberg-Chandrasekhar core mass fraction strongly analogous to the classic numerical result for an isothermal core, a radiative envelope and a μ-jump (i.e. a molecular weight jump) at the interface. Our model consists of an nc = 5 core, an ne = 1 envelope and a μ-jump by a factor ≥ 3; the core mass fraction cannot exceed 2/π. We use the classic U, V plane to show that composite models will exhibit a Schönberg-Chandrasekhar limit only if the core is 'soft', i.e. has nc ≥ 5, and the envelope is 'hard', i.e. has nc < 5; in the critical case (nc = 5), the limit only exists if the μ-jump is sufficiently large, ≥ 6/(ne + 1)."

This paper uses analytic techniques to derive precisely the same sequence of $(n_c,n_e)=(5,1)$ bipolytropic models that we have presented above.

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